Question

Flexibility coefficients are used in which of the following method?

• A. Force method
• B. Displacement method
• C. Both force and displacement method
• D. Virtual force method

Hint:

Remember: Force method → flexibility coefficients, Displacement method → stiffness coefficients.

Answer 1

The Correct Option is A

Step 1: Understanding flexibility coefficients.
Flexibility coefficients represent the displacement produced at a point due to a unit force applied at another point.
Step 2: Relation with structural analysis methods.
In the force method (also known as the flexibility method), unknown forces are taken as redundants and compatibility conditions are written using flexibility coefficients.
Step 3: Analysis of options.
(A) Force method: Correct, flexibility coefficients are fundamental to this method.
(B) Displacement method: Incorrect, this method primarily uses stiffness coefficients.
(C) Both force and displacement method: Incorrect, as each method uses different coefficients.
(D) Virtual force method: Incorrect, this method is mainly used for deflection calculations.
Step 4: Conclusion.
Hence, flexibility coefficients are used in the force method.

Answer 2

Another way to confirm this is to look at how the flexibility matrix relates to the two families of structural analysis methods through its role as the mathematical inverse of the stiffness matrix, rather than only stating where flexibility coefficients are used from definition.

In matrix structural analysis, the stiffness matrix \([K]\) relates forces to displacements as \([F] = [K][\delta]\), while the flexibility matrix \([f]\) is its inverse, \([f] = [K]^{-1}\), and relates displacements to forces as \([\delta] = [f][F]\). Whichever unknowns a method solves for directly determines which of these two matrices is the natural one to use.

1. Force method: Here the unknowns are redundant forces, and compatibility equations are written in terms of displacements caused by these unknown forces — which is exactly the input-output relationship the flexibility matrix expresses (force in, displacement out). So flexibility coefficients are the natural tool for this method.
2. Displacement method: Here the unknowns are joint displacements, and equilibrium equations are written in terms of forces caused by these displacements — the relationship \([F]=[K][\delta]\), which is the stiffness matrix, not its inverse. So flexibility coefficients are not the natural tool here.
3. Both force and displacement method: Since the two methods solve for different types of unknowns (forces vs. displacements) and therefore require inverse matrices of each other, they cannot both rely on the same flexibility formulation.
4. Virtual force method: This technique is used for computing a single deflection value using virtual work, not for setting up the system of simultaneous equations that defines the force method of full structural analysis, so it is not the method being referred to here.

Since the force method's unknowns are forces and its governing equations use the force-to-displacement relationship, it is naturally built on the flexibility matrix.

Therefore, the correct answer is Force method.